Copyright	(c) Andy Gill 2001, (c) Oregon Graduate Institute of Science and Technology, 2001
License	BSD-style (see the file libraries/base/LICENSE)
Maintainer	libraries@haskell.org
Stability	experimental
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Monoid

Contents

Monoid typeclass
Bool wrappers
Num wrappers
Maybe wrappers
Alternative wrapper

Description

A class for monoids (types with an associative binary operation that has an identity) with various general-purpose instances.

Synopsis

class Monoid a where
- mempty :: a
- mappend :: a -> a -> a
- mconcat :: [a] -> a
(<>) :: Monoid m => m -> m -> m
newtype Dual a = Dual {
- getDual :: a
}
newtype Endo a = Endo {
- appEndo :: a -> a
}
newtype All = All {
- getAll :: Bool
}
newtype Any = Any {
- getAny :: Bool
}
newtype Sum a = Sum {
- getSum :: a
}
newtype Product a = Product {
- getProduct :: a
}
newtype First a = First {
- getFirst :: Maybe a
}
newtype Last a = Last {
- getLast :: Maybe a
}
newtype Alt f a = Alt {
- getAlt :: f a
}

`Monoid` typeclass

class Monoid a where Source

The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following laws:

```
mappend mempty x = x
```
```
mappend x mempty = x
```

mappend x (mappend y z) = mappend (mappend x y) z

```
mconcat = foldr  mappend mempty
```

The method names refer to the monoid of lists under concatenation, but there are many other instances.

Some types can be viewed as a monoid in more than one way, e.g. both addition and multiplication on numbers. In such cases we often define newtypes and make those instances of Monoid , e.g. Sum and Product.

Minimal complete definition

mempty, mappend

Methods

mempty :: a Source

Identity of mappend

mappend :: a -> a -> a Source

An associative operation

mconcat :: [a] -> a Source

Fold a list using the monoid. For most types, the default definition for mconcat will be used, but the function is included in the class definition so that an optimized version can be provided for specific types.

Instances

Monoid Ordering Source

Methods

mempty :: Ordering Source

mappend :: Ordering -> Ordering -> Ordering Source

mconcat :: [Ordering] -> Ordering Source

Monoid () Source

Methods

mempty :: () Source

mappend :: () -> () -> () Source

mconcat :: [()] -> () Source

Monoid Any Source

Methods

mempty :: Any Source

mappend :: Any -> Any -> Any Source

mconcat :: [Any] -> Any Source

Monoid All Source

Methods

mempty :: All Source

mappend :: All -> All -> All Source

mconcat :: [All] -> All Source

Monoid Lifetime Source

mappend == elSupremum

Methods

mempty :: Lifetime Source

mappend :: Lifetime -> Lifetime -> Lifetime Source

mconcat :: [Lifetime] -> Lifetime Source

Monoid Event Source

Methods

mempty :: Event Source

mappend :: Event -> Event -> Event Source

mconcat :: [Event] -> Event Source

Monoid [a] Source

Methods

mempty :: [a] Source

mappend :: [a] -> [a] -> [a] Source

mconcat :: [[a]] -> [a] Source

Monoid a => Monoid (Maybe a) Source

Lift a semigroup into Maybe forming a Monoid according to http://en.wikipedia.org/wiki/Monoid: "Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e*e = e and e*s = s = s*e for all s ∈ S." Since there is no "Semigroup" typeclass providing just mappend , we use Monoid instead.

Methods

mempty :: Maybe a Source

mappend :: Maybe a -> Maybe a -> Maybe a Source

mconcat :: [Maybe a] -> Maybe a Source

Monoid (Last a) Source

Methods

mempty :: Last a Source

mappend :: Last a -> Last a -> Last a Source

mconcat :: [Last a] -> Last a Source

Monoid (First a) Source

Methods

mempty :: First a Source

mappend :: First a -> First a -> First a Source

mconcat :: [First a] -> First a Source

Num a => Monoid (Product a) Source

Methods

mempty :: Product a Source

mappend :: Product a -> Product a -> Product a Source

mconcat :: [Product a] -> Product a Source

Num a => Monoid (Sum a) Source

Methods

mempty :: Sum a Source

mappend :: Sum a -> Sum a -> Sum a Source

mconcat :: [Sum a] -> Sum a Source

Monoid (Endo a) Source

Methods

mempty :: Endo a Source

mappend :: Endo a -> Endo a -> Endo a Source

mconcat :: [Endo a] -> Endo a Source

Monoid a => Monoid (Dual a) Source

Methods

mempty :: Dual a Source

mappend :: Dual a -> Dual a -> Dual a Source

mconcat :: [Dual a] -> Dual a Source

Monoid b => Monoid (a -> b) Source

Methods

mempty :: a -> b Source

mappend :: (a -> b) -> (a -> b) -> a -> b Source

mconcat :: [a -> b] -> a -> b Source

(Monoid a, Monoid b) => Monoid (a, b) Source

Methods

mempty :: (a, b) Source

mappend :: (a, b) -> (a, b) -> (a, b) Source

mconcat :: [(a, b)] -> (a, b) Source

Monoid (Proxy k s) Source

Methods

mempty :: Proxy k s Source

mappend :: Proxy k s -> Proxy k s -> Proxy k s Source

mconcat :: [Proxy k s] -> Proxy k s Source

Monoid a => Monoid (Const a b) Source

Methods

mempty :: Const a b Source

mappend :: Const a b -> Const a b -> Const a b Source

mconcat :: [Const a b] -> Const a b Source

(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) Source

Methods

mempty :: (a, b, c) Source

mappend :: (a, b, c) -> (a, b, c) -> (a, b, c) Source

mconcat :: [(a, b, c)] -> (a, b, c) Source

Alternative f => Monoid (Alt * f a) Source

Methods

mempty :: Alt * f a Source

mappend :: Alt * f a -> Alt * f a -> Alt * f a Source

mconcat :: [Alt * f a] -> Alt * f a Source

(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d) Source

Methods

mempty :: (a, b, c, d) Source

mappend :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source

mconcat :: [(a, b, c, d)] -> (a, b, c, d) Source

(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e) Source

Methods

mempty :: (a, b, c, d, e) Source

mappend :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source

mconcat :: [(a, b, c, d, e)] -> (a, b, c, d, e) Source

(<>) :: Monoid m => m -> m -> m infixr 6 Source

An infix synonym for mappend .

Since: 4.5.0.0

newtype Dual a Source

The dual of a Monoid , obtained by swapping the arguments of mappend .

Constructors

Dual

Fields

getDual :: a

Instances

Generic1 Dual Source

Associated Types

type Rep1 (Dual :: * -> *) :: * -> * Source

Methods

from1 :: Dual a -> Rep1 Dual a Source

to1 :: Rep1 Dual a -> Dual a Source

Bounded a => Bounded (Dual a) Source

Methods

minBound :: Dual a Source

maxBound :: Dual a Source

Eq a => Eq (Dual a) Source

Methods

(==) :: Dual a -> Dual a -> Bool

(/=) :: Dual a -> Dual a -> Bool

Ord a => Ord (Dual a) Source

Methods

compare :: Dual a -> Dual a -> Ordering

(<) :: Dual a -> Dual a -> Bool

(<=) :: Dual a -> Dual a -> Bool

(>) :: Dual a -> Dual a -> Bool

(>=) :: Dual a -> Dual a -> Bool

max :: Dual a -> Dual a -> Dual a

min :: Dual a -> Dual a -> Dual a

Read a => Read (Dual a) Source

Methods

readsPrec :: Int -> ReadS (Dual a) Source

readList :: ReadS [Dual a] Source

readPrec :: ReadPrec (Dual a) Source

readListPrec :: ReadPrec [Dual a] Source

Show a => Show (Dual a) Source

Methods

showsPrec :: Int -> Dual a -> ShowS Source

show :: Dual a -> String Source

showList :: [Dual a] -> ShowS Source

Generic (Dual a) Source

Associated Types

type Rep (Dual a) :: * -> * Source

Methods

from :: Dual a -> Rep (Dual a) x Source

to :: Rep (Dual a) x -> Dual a Source

Monoid a => Monoid (Dual a) Source

Methods

mempty :: Dual a Source

mappend :: Dual a -> Dual a -> Dual a Source

mconcat :: [Dual a] -> Dual a Source

type Rep1 Dual Source

type Rep (Dual a) Source

newtype Endo a Source

The monoid of endomorphisms under composition.

Constructors

Endo

Fields

appEndo :: a -> a

Instances

Generic (Endo a) Source

Associated Types

type Rep (Endo a) :: * -> * Source

Methods

from :: Endo a -> Rep (Endo a) x Source

to :: Rep (Endo a) x -> Endo a Source

Monoid (Endo a) Source

Methods

mempty :: Endo a Source

mappend :: Endo a -> Endo a -> Endo a Source

mconcat :: [Endo a] -> Endo a Source

type Rep (Endo a) Source

`Bool` wrappers

newtype All Source

Boolean monoid under conjunction (&& ).

Constructors

All

Fields

getAll :: Bool

Instances

Bounded All Source

Methods

minBound :: All Source

maxBound :: All Source

Eq All Source

Methods

(==) :: All -> All -> Bool

(/=) :: All -> All -> Bool

Ord All Source

Methods

compare :: All -> All -> Ordering

(<) :: All -> All -> Bool

(<=) :: All -> All -> Bool

(>) :: All -> All -> Bool

(>=) :: All -> All -> Bool

max :: All -> All -> All

min :: All -> All -> All

Read All Source

Methods

readsPrec :: Int -> ReadS All Source

readList :: ReadS [All] Source

readPrec :: ReadPrec All Source

readListPrec :: ReadPrec [All] Source

Show All Source

Methods

showsPrec :: Int -> All -> ShowS Source

show :: All -> String Source

showList :: [All] -> ShowS Source

Generic All Source

Associated Types

type Rep All :: * -> * Source

Methods

from :: All -> Rep All x Source

to :: Rep All x -> All Source

Monoid All Source

Methods

mempty :: All Source

mappend :: All -> All -> All Source

mconcat :: [All] -> All Source

type Rep All Source

newtype Any Source

Boolean monoid under disjunction (|| ).

Constructors

Any

Fields

getAny :: Bool

Instances

Bounded Any Source

Methods

minBound :: Any Source

maxBound :: Any Source

Eq Any Source

Methods

(==) :: Any -> Any -> Bool

(/=) :: Any -> Any -> Bool

Ord Any Source

Methods

compare :: Any -> Any -> Ordering

(<) :: Any -> Any -> Bool

(<=) :: Any -> Any -> Bool

(>) :: Any -> Any -> Bool

(>=) :: Any -> Any -> Bool

max :: Any -> Any -> Any

min :: Any -> Any -> Any

Read Any Source

Methods

readsPrec :: Int -> ReadS Any Source

readList :: ReadS [Any] Source

readPrec :: ReadPrec Any Source

readListPrec :: ReadPrec [Any] Source

Show Any Source

Methods

showsPrec :: Int -> Any -> ShowS Source

show :: Any -> String Source

showList :: [Any] -> ShowS Source

Generic Any Source

Associated Types

type Rep Any :: * -> * Source

Methods

from :: Any -> Rep Any x Source

to :: Rep Any x -> Any Source

Monoid Any Source

Methods

mempty :: Any Source

mappend :: Any -> Any -> Any Source

mconcat :: [Any] -> Any Source

type Rep Any Source

`Num` wrappers

newtype Sum a Source

Monoid under addition.

Constructors

Sum

Fields

getSum :: a

Instances

Generic1 Sum Source

Associated Types

type Rep1 (Sum :: * -> *) :: * -> * Source

Methods

from1 :: Sum a -> Rep1 Sum a Source

to1 :: Rep1 Sum a -> Sum a Source

Bounded a => Bounded (Sum a) Source

Methods

minBound :: Sum a Source

maxBound :: Sum a Source

Eq a => Eq (Sum a) Source

Methods

(==) :: Sum a -> Sum a -> Bool

(/=) :: Sum a -> Sum a -> Bool

Num a => Num (Sum a) Source

Methods

(+) :: Sum a -> Sum a -> Sum a Source

(-) :: Sum a -> Sum a -> Sum a Source

(*) :: Sum a -> Sum a -> Sum a Source

negate :: Sum a -> Sum a Source

abs :: Sum a -> Sum a Source

signum :: Sum a -> Sum a Source

fromInteger :: Integer -> Sum a Source

Ord a => Ord (Sum a) Source

Methods

compare :: Sum a -> Sum a -> Ordering

(<) :: Sum a -> Sum a -> Bool

(<=) :: Sum a -> Sum a -> Bool

(>) :: Sum a -> Sum a -> Bool

(>=) :: Sum a -> Sum a -> Bool

max :: Sum a -> Sum a -> Sum a

min :: Sum a -> Sum a -> Sum a

Read a => Read (Sum a) Source

Methods

readsPrec :: Int -> ReadS (Sum a) Source

readList :: ReadS [Sum a] Source

readPrec :: ReadPrec (Sum a) Source

readListPrec :: ReadPrec [Sum a] Source

Show a => Show (Sum a) Source

Methods

showsPrec :: Int -> Sum a -> ShowS Source

show :: Sum a -> String Source

showList :: [Sum a] -> ShowS Source

Generic (Sum a) Source

Associated Types

type Rep (Sum a) :: * -> * Source

Methods

from :: Sum a -> Rep (Sum a) x Source

to :: Rep (Sum a) x -> Sum a Source

Num a => Monoid (Sum a) Source

Methods

mempty :: Sum a Source

mappend :: Sum a -> Sum a -> Sum a Source

mconcat :: [Sum a] -> Sum a Source

type Rep1 Sum Source

type Rep (Sum a) Source

newtype Product a Source

Monoid under multiplication.

Constructors

Product

Fields

getProduct :: a

Instances

Generic1 Product Source

Associated Types

type Rep1 (Product :: * -> *) :: * -> * Source

Methods

from1 :: Product a -> Rep1 Product a Source

to1 :: Rep1 Product a -> Product a Source

Bounded a => Bounded (Product a) Source

Methods

minBound :: Product a Source

maxBound :: Product a Source

Eq a => Eq (Product a) Source

Methods

(==) :: Product a -> Product a -> Bool

(/=) :: Product a -> Product a -> Bool

Num a => Num (Product a) Source

Methods

(+) :: Product a -> Product a -> Product a Source

(-) :: Product a -> Product a -> Product a Source

(*) :: Product a -> Product a -> Product a Source

negate :: Product a -> Product a Source

abs :: Product a -> Product a Source

signum :: Product a -> Product a Source

fromInteger :: Integer -> Product a Source

Ord a => Ord (Product a) Source

Methods

compare :: Product a -> Product a -> Ordering

(<) :: Product a -> Product a -> Bool

(<=) :: Product a -> Product a -> Bool

(>) :: Product a -> Product a -> Bool

(>=) :: Product a -> Product a -> Bool

max :: Product a -> Product a -> Product a

min :: Product a -> Product a -> Product a

Read a => Read (Product a) Source

Methods

readsPrec :: Int -> ReadS (Product a) Source

readList :: ReadS [Product a] Source

readPrec :: ReadPrec (Product a) Source

readListPrec :: ReadPrec [Product a] Source

Show a => Show (Product a) Source

Methods

showsPrec :: Int -> Product a -> ShowS Source

show :: Product a -> String Source

showList :: [Product a] -> ShowS Source

Generic (Product a) Source

Associated Types

type Rep (Product a) :: * -> * Source

Methods

from :: Product a -> Rep (Product a) x Source

to :: Rep (Product a) x -> Product a Source

Num a => Monoid (Product a) Source

Methods

mempty :: Product a Source

mappend :: Product a -> Product a -> Product a Source

mconcat :: [Product a] -> Product a Source

type Rep1 Product Source

type Rep (Product a) Source

`Maybe` wrappers

To implement find or findLast on any Foldable:

findLast :: Foldable t => (a -> Bool) -> t a -> Maybe a
findLast pred = getLast . foldMap (x -> if pred x
 then Last (Just x)
 else Last Nothing)

Much of Data.Map's interface can be implemented with Data.Map.alter. Some of the rest can be implemented with a new alterA function and either First or Last :

alterA :: (Applicative f, Ord k) =>
 (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)
instance Monoid a => Applicative ((,) a) -- from Control.Applicative

insertLookupWithKey :: Ord k => (k -> v -> v -> v) -> k -> v
 -> Map k v -> (Maybe v, Map k v)
insertLookupWithKey combine key value =
 Arrow.first getFirst . alterA doChange key
 where
 doChange Nothing = (First Nothing, Just value)
 doChange (Just oldValue) =
 (First (Just oldValue),
 Just (combine key value oldValue))

newtype First a Source

Maybe monoid returning the leftmost non-Nothing value.

First a is isomorphic to Alt Maybe a, but precedes it historically.

Constructors

First

Fields